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Proof Theory and Proof Mining II: Formal Theories of Analysis

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Proof Theory and Proof Mining II: Formal Theories of Analysis Proof Theory and Proof Mining II: Formal Theories of Analysis Jeremy Avigad Department of Philosophy and Department of Mathematical Sciences Carnegie Mellon University July 2015 Sequence of Topics 1. Computable Analysis 2. Formal Theories of Analysis 3. The Dialectica Interpretation and Applications 4. Ultraproducts and Nonstandard Analysis Formal theories of analysis Analysis can be formalized in set theory. But Part I showed that we don't always need big sets: many objects of analysis can be represented by sets of natural numbers. Hilbert and Bernays' Grundlagen der Mathematik, volume II, 1936, showed that portions of analysis can be formalized in second-order arithmetic (with third-order parameters). SOA is often called \analysis" for that reason. Kreisel: provability in restricted theories provides additional information. Axiomatic theories We will consider: • Primitive recursive arithmetic (PRA) • First-order arithmetic (PA; HA) (and subsystems) • Second-order arithmetic (PA2 , HA2 ) (and subsystems) These come in classical and intuitionistic versions. We will also consider theories of finite type. Primitive recursive arithmetic Recall that the primitive recursive functions include • zero, 0 • successor, S n • projections, pi (x1;:::; xn) = xi and are closed under • composition: f (~x) = h(g1(~x);:::; gn(~x)) • primitive recursion: f (0; ~z) = g(~z); f (x + 1; ~z) = h(f (x; ~z); x; ~z) Primitive recursive arithmetic Primitive recursive arithmetic is an axiomatic theory, with • 0 6= S(x), S(x) = S(y) ! x = y • defining equations for the primitive recursive functions • quantifier-free induction: '(0) '(x) ! '(x + 1) '(t) In PRA, can handle “finitary" proofs. Primitive recursive arithmetic Just as all \reasonable" computable functions are primitive recursive, all \reasonable" facts about them can be proved in PRA. In fact, it is surprisingly hard to find ordinary mathematical theorems that can be stated in the language of PRA but not proved there. Primitive recursive arithmetic PRA can be presented either as a first-order theory (classical or intuitionistic) or as a quantifier-free calculus. Theorem. Suppose first-order classical PRA proves 8x 9y '(x; y), with ' quantifier-free. Then for some function symbol f , quantifier-free PRA proves '(x; f (x)). Proof. First, show PRA has a set of universal axioms. Define bounded quantification, and replace induction by 8y ('(0) ^ 8x < y ('(x) ! '(S(x))) ! '(y)): Then apply Herbrand's theorem. Herbrand's theorem Theorem. Suppose T is a universal axiomatized first-order theory, and T ` 9~x '(~x), ' quantifier free. Then there are sequences of terms ~t1;:::~tn and a quantifier-free proof of '(~t1) _ ::: _ '(~tn) from instances of axioms of T . Proof. WLOG, assume T has a constant symbol. If the conclusion fails, T [ f:'(~t)g is consistent, where ~t ranges over all tuples of terms. Let M be a model. Let M0 be the submodel with universe ftMg. Then M0 j= T [ f8~x :'(~x)g. First-order arithmetic First-order arithmetic is essentially PRA plus induction. Peano arithmetic (PA) is classical, Heyting arithmetic (HA) is intuitionistic. Language: 0; S; +; ×; <. Axioms: quantifier-free defining axioms, induction. A formula is • ∆0 if every quantifier is bounded • Σ1 if of the form 9~x ', ' 2 ∆0 • Π1 if of the form 8~x ', ' 2 ∆0 • ∆1 if equivalent to Σ1 and Π1 Primitive recursive functions / relations have Σ1=∆1 definitions. First-order arithmetic IΣ1 is the restriction of PA with induction for only Σ1 formulas. This theory suffices to define the primitive recursive functions, and hence interpret PRA. Conversely: Theorem (Parsons, Mints, Takeuti). IΣ1 is conservative over PRA for Π2 sentences: if IΣ1 ` 8x 9y '(x; y); with ' quantifier-free, then PRA ` '(x; f (x)) for some function symbol f . Conservativity of IΣ1 over PRA There are various ways to prove this theorem. Syntactic proofs: • Using cut elimination or normalization. • Using the Dialectica interpretation (plus normalization). Model-theoretic proofs: • A model-theoretic argument due to Friedman. • A \saturation" construction. Double-negation translations The G¨odel-Gentzendouble-negation translation interprets classical logic in minimal logic: • AN ≡ ::A for atomic A • (' _ )N ≡ :(:'N ^ : N ) • (9x ')N ≡ :8x :'N . The translation commutes with 8, ^, !. Theorem. If Γ ` ' classically, ΓN ` 'N in minimal logic. The proof uses induction on derivations. For example, (' _:')N is easily proved. Reducing PA to HA Theorem. If PA proves a formula, ', then HA proves 'N . Proof. For each axiom of ' of PA, HA proves 'N . Corollary. If PA proves 8x 9y R(x; y), where R is primitive recursive, then HA proves 8x :8y :R(x; y). Better theorem. If PA proves 8x 9y R(x; y), where R is primitive recursive, then HA proves 8x 9y R(x; y). There are various ways to prove this; later, we will use the Dialectica interpretation. Second order arithmetic The language is two-sorted: • variables x; y; z;::: and functions 0; S; +; × on one sort • variables X ; Y ; Z;::: on the other sort • a relation t 2 X between the two sorts Axioms: • axioms of PA, with induction extended to the bigger language • comprehension: 9X 8y (y 2 X $ '(y; ~z)) The \standard model" is (N; P(N);:::), but there are smaller ones. An !-model is a model where the first-order part is standard, i.e. N. Second order arithmetic We can define equality on the second sort by X = Y := 8z (z 2 X $ z 2 Y ): The usual axioms of equality (including substitution) follow. This is known as \extensional equality." Alternatively, we can take equality as a basic symbol, and add the axiom above. This second system is interpreted in the first. Second order arithmetic Functions vs. sets: • With functions basic, interpret sets as characteristic functions: x 2 S ≡ χS (x) = 1: • With relations basic, interpret functions as functional relations, 8x 9!y R(x; y) Can add choice axioms: 8x 9y '(x; y) ! 9f 8y '(x; f (x)): Second order arithmetic From a proof-theoretic perspective, second-order arithmetic is very strong. We obtain weaker systems by: • restricting comprehension • restricting induction Subsystems of second-order arithmetic The big five: 0 • RCA0 : recursive (∆1) comprehension (formalized computable analysis) • WKL0 : weak K¨onig'slemma (a form of compactness) • ACA0 : arithmetic comprehension (analytic principles like the least-upper bound principle.) • ATR0 : transfinitely iterated arithmetic comprehension (transfinite constructions) 1 1 • Π1 −CA0 :Π1 comprehension (strong analytic principles) We will focus on the first three. RCA0 The axioms of RCA0 are as follows: • quantifier-free axioms for 0; S; +; ×; < • induction, restricted to Σ1 formulas (with both number and set parameters): '(0) ^ 8x ('(x) ! '(x + 1)) ! 8x '(x) • the recursive comprehension axiom, (RCA): 8x ('(x) $ (x)) ! 9Y 8x (x 2 Y $ '(x)) where ' is Σ1 and is Π1. RCA0 Notice that the induction schema includes set induction: 0 2 Y ^ 8x (x 2 Y ! x + 1 2 Y ) ! 8x (x 2 Y ): It is slightly stronger. Since RCA0 includes IΣ1 , we can act as though primitive recursive arithmetic is \built-in." RCA0 (RCA) says that if a set exists if it has a computably enumerable definition as well as a co-computably enumerable definition (relative to others sets in the universe). Roughly, it allows you to define computable sets and relations. Let REC denote the set of recursive sets. Then (N; REC;:::) is the minimal !-model. Analysis in RCA0 is roughly \formalized computable analysis." WKL0 Remember that a tree on f0; 1g is a set T of finite binary sequences closed under initial segments: Tree(T ) ≡ 8σ; τ (σ 2 T ^ τ ⊆ σ ! τ 2 T ): We can say T is infinite as follows: Infinite(T ) ≡ 8n 9σ (σ 2 T ^ length(σ) = n): A set P is a path through T if, when viewed as an infinite binary sequence, every initial segments is in T : Path(P; T ) ≡ 8σ ((8i < length(σ) ; i 2 P $ (σ)i = 1) ! σ 2 T ): WKL0 Weak K¨onig'slemma (WKL) says that every infinite binary tree has a path: 8T (Tree(T ) ^ Infinite(T ) ! 9P Path(P; T )): The theory WKL0 is RCA0 + (WKL). WKL0 Since there are computable infinite binary trees with no computable path, REC is not an !-model. It has a model where every set is computable from 00. Remarkably, (WKL) has no minimal !-model, but the intersection of all !-models is REC. WKL0 Over RCA0 ,(WKL) is equivalent to each of these: • the Heine-Borel theorem (for [0; 1]) • Every open cover of f0; 1g! has a finite subcover. • Every continuous function on [0; 1] is uniformly continuous • Every continuous function on [0; 1] is bounded Here, [0; 1] can be replaced by any compact space. (More on this below.) ACA0 A formula is arithmetic if it has only first-order quantifiers. ACA0 adds to RCA0 the arithmetic comprehsion axiom,(ACA): 9Y 8x (x 2 Y $ '(x)) where ' is arithmetic (possibly with number and set parameters). The smallest !-model is the collection of arithmetically definable sets. Notice that set-induction now gives us induction for every arithmetic formula. ACA0 Theorem. Over RCA0 ,(ACA) is equivalent to each of these: • Every bounded increasing sequence of real numbers has a least upper bound. • Every bounded sequence of real nubmers has a least upper bound. • Every bounded sequence of real numbers has a convergence subsequence. • Every sequence of points in a compact metric space has a convergent subsequence. Metamathematics of SOSOA We have considered subsystems of second-order arithmetic from the point of view of: • their minimal models • what they can be prove Knowing that a mathematical theorem is provable in a restricted theory, T , provides additional information. Conservation results Many central proof theoretic results have the following form: 0 For any ' 2 Γ, if T1 ` ', then T2 ` ' . These can be used to reduce • infinitary theories to finitary ones • classical theories to constructive ones • impredicative theories to predicative ones • nonstandard theories to standard ones • higher-order theories to first-order ones • first-order theories to quantifier-free ones Conservation results From a foundational point of view, conservations results explain or interpret theories that are more • abstract, • mysterious, or • dubious, in terms of ones that are more • concrete, • familiar, or • trustworthy.
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